Standard methods for meta-analysis of dose–response data in epidemiology assume a model with a single scalar parameter, such as log-linear relationships between exposure and outcome; such models are implicitly unbounded. In contrast, in pharmacology, multi-parameter models, such as the widely used Emax model, are used to describe relationships that are bounded above and below. We propose methods for estimating the parameters of a dose–response model by meta-analysis of summary data from the results of randomized controlled trials of a drug, in which each trial uses multiple doses of the drug of interest (possibly including dose 0 or placebo). We assume that, for each randomized arm of each trial, the mean and standard error of a continuous response measure and the corresponding allocated dose are available. We consider weighted least squares fitting of the model to the mean and dose pairs from all arms of all studies, and a two-stage procedure in which scalar inverse-variance meta-analysis is performed at each dose, and the dose–response model is fitted to the results by weighted least squares. We then compare these with two further methods inspired by network meta-analysis that fit the model to the contrasts between doses. We illustrate the methods by estimating the parameters of the Emax model to a collection of multi-arm, multiple-dose, randomized controlled trials of alogliptin, a drug for the management of diabetes mellitus, and further examine the properties of the four methods with sensitivity analyses and a simulation study. We find that all four methods produce broadly comparable point estimates for the parameters of most interest, but a single-stage method based on contrasts between doses produces the most appropriate confidence intervals. Although simpler methods may have pragmatic advantages, such as the use of standard software for scalar meta-analysis, more sophisticated methods are nevertheless preferable for their advantages in estimation.
Methods for meta-analysis of pharmacodynamic dose–response data with application to multi-arm studies of alogliptin